The Theoretic Arithmetic of the Pythagoreans

✍ Scribed by Thomas Taylor, Manly P. Hall

• Publisher: Samuel Weiser, Inc.
• Year: 1983
• Tongue: English
• Leaves: 311
• Category: Library

✦ Synopsis

This book contains the essential elements of all that has been written on the subject by such men as Iamblichus, Boetius and Nicomachus together with some particulars respecting other numbers, which are not found in the writings of any ancient or modern mathematicians. Also the work concerns itself with the Pythagorean philosophy of numbers, and the development of their mystical and theological arithmetic.

In his impressive introduction, Manly P. Hall points out that while too little remains of the system of numerical divination, there are fragments of genuine Pythagorean Arithmetic to be found, in which he held that through the study of mathematics all men are able to commune with the gods. Manly P. Hall continues by showing that Numerology was perfected by Pythagoras as a system of divination. Pythagoras proved the worth of his system by using it successfully.

This volume contains three books. Book One opens on the basic principle that Arithmetic has priority over other mathematical disciplines, and proves its theory by diagrams. It also records that the motion of the stars is accompanied with harmonic modulations. Book Two proceeds to show all inequality may be reduced to equality. Book Three shows clearly the manner in which Pythagoreans philosophized about the value of numbers.

Thomas Taylor, author of this excellent treatise, was the greatest Platonist of the modern world. Taylor believed the Platonic philosophy was a workable philosophy of life. He dedicated his life to translating the Greek classics. Among his better known works are Aristotle, Plotinus and Euclid. Taylor himself was called “The Modern Plato.”

✦ Table of Contents

Cover
Frontispiece
Half title
Imprint
INTRODUCTION By MANLY P. HALL
THEORETIC ARITHMETIC
INTRODUCTION
BOOK ONE
I. On the priority of Arithmetic to the other mathematical disciplines
II. On the definition of number and the monad
III. On the division of numbers, and the various definitions of the even and the odd
IV. On the predominance of the monad
V. The division of the even number — And on the evenly-even number, and its properties
VI. On the evenly-odd number, and its properties
VII. On the unevenly-even number, and its properties, etc.
VIII. On the odd number, and its division
IX. On the first and incomposite number
X. On the second and composite number
XIII. On the method of discovering the commensurability, or incommensurability of these numbers
XIV. Another division of the even number according to the perfect, deficient, and superperfect, or superabundant
XV. On the generation of the perfect number, and its similitude to virtue
XVI. On relative quantity, and the species of greater and less inequality
XVII. On multiple inequality, its species, and the generation of them
XVIII. On the superparticular number, its species, and the generation of them, etc.
XIX. That the multiple 1s more ancient than the other species of inequality
XX. On the third species of inequality, which is called superpartient
XXI. On the multiple superparticular and. superpartient ratio
XXII. A demonstration that all ineqality proceeds from equality
BOOK TWO
I. How all inequality may be reduced to equality
II. On discovering in each number, how many numbers of the same ratio may precede—description of them and an exposition of the description
III. The method of finding the superparticular intervals from which the multiple interval is produced
IV. On the quantity subsisting by itself, which is considered in geometrical figures, etc.
V. On plane rectilinear figures.—That the triangle is the principle of them
VI. On square numbers, their sides, and generation
VII. On pentagons, their sides, and generation
VIII. On hexagons and heptagons, and the generation of them
IX. On the figurate numbers that are produced from figurate numbers; and that the triangular number is the principle of all the rest
X. On solid numbers.—On the pyramid, that it is the principle of all solid figures, in the same manner as the triangle, of plane figures; and on its species
XI. The generation of solid numbers
XII. On defective pyramids
XIII. On the numbers called cubes, wedges, and parallelopipedons
XIV. On the numbers called HETEROMEKEIS, Of, LONGER IN THE OTHER PART;—and on oblong numbers, and the generation of them
XV. That squares are generated from odd numbers; but the HETEROMEKEIS from even numbers
XVI. On the generation of the numbers called LATERCULI or tyles, and of those denominated asseres or planks
XVII. On the nature of sameness and difference, and what the numbers are which participate of these
XVIII. That all things consist of sameness and difference; and that the truth of this is primarily to be seen in numbers
XIX. That from the nature of the numbers which are characterized by sameness, and from the nature of those which are characterized by difference, viz. from squares, and numbers LONGER IN THE OTHER PART, all the habitudes of proportions consist
XX. That from squares and numbers LONGER IN THE OTHER PART, all numerical figures consist—How numbers LONGER IN THE OTHER PART, are produced from squares and vice versa, the latter from the former, etc.
XXI. What agreement there 1s in difference and in ratio, between squares and numbers LONGER IN THE OTHER PART, when they are alternately arranged
XXII. A demonstration that squares and cubes partake of the nature of sameness
XXIII. On proportionality, or analogy
XXIV. On the proportionality which was known to the ancients, and what the proportions are which those posterior to them have added.—And on arithmetical proportionality, and its properties
XXV. On the geometrical middle and its properties
XXVI. That plane numbers are conjoined by one medium only, but solid numbers by two media
XXVII. On the harmonic middle and its properties
XXVIII. Why this middle 1s called harmonic, and on geometrical harmony
XXIX. How two terms being constituted on either side, the arithmetic, geometric, and harmonic middle 1s alternately changed between them; and on the generation of them
XXX. On the three middles which are contrary to the harmonic and geometric middles
XXXI. On the four middles which the ancients posterior. to those before mentioned, invented for the purpose of giving completion to the decad
XXXII. On the greatest and most perfect symphony, which is extended in three intervals;—and also on the less symphony
XXXIII. On amicable numbers
XXXIV. On lateral and diametrical numbers.
XXXV. On arithmetical and geometrical series
XXXVI. On imperfectly amıcable numbers
XXXVII. On the series of unevenly-even numbers
XXXVIII. On the aggregate of the parts of the terms of different series
XXXIX. On the series of terms arising from the multiplication of evenly-even numbers, by the sums produced by the addition of them
XL. On another species of imperfectly amicable numbers
XLI. On the geometric number, in the eighth book of the Republic of Plato
BOOK THREE
I. On the manner in which the Pythagoreans philosophized about numbers
II. On mathematical and physical number
III. On the monad
IV. On the duad
V. On the triad
VI. On the tetrad
VII. On the pentad
VIII. On the hexad
IX. On the heptad
X. On the ogdoad
XI. On the ennead
XII. On the decad
XIII. On the properties of the monad
XIV. On the properties of the duad, and triad
XV. On the properties of the tetrad, pentad, and hexad
XVI. On the properties of the hebdomad
XVII. On the ogdoad, ennead, and decad
XVIII. Additional observations on numbers
ADDITIONAL NOTES

✦ Subjects

pythagoreanism;theoreticarithme0000tayl